If f(x) is continuous on a closed interval [a, b], then at some point c in the interval [a, b] the
following is true:f(x) dx = f(c)(b − a)This tells you that the area under the curve of f(x) on the interval [a, b] is equal to the value of the function
at some value c (between a and b) times the length of the interval. If you look at this graphically, you can
see that you’re finding the area of a rectangle whose base is the interval and whose height is some value
of f(x) that creates a rectangle with the same area as the area under the curve.
The number f(c) gives us the average value of f on [a, b]. Thus, if we rearrange the theorem, we get the
formula for finding the average value of f(x) on [a, b].
f(c) = f(x) dxThere’s all you need to know about finding average values. Try some examples.